Abstract

With the help of the results derived in Part I of this investigation, a new representation is obtained for the field arising from diffraction of a monochromatic wave by an aperture in an opaque screen. It is shown that within the accuracy of the Kirchhoff diffraction theory, the diffracted field UK(P) may be expressed under very general conditions in the form

UK(P)=U(B)(P)+jFj(P).

Here U(B) represents a disturbance originating at each point of the boundary of the aperture and ΣjFj(P) represents the total effect of disturbances propagated from certain special points Qj in the aperture. Expressions for U(B) and Fj(P) are given, in terms of the new vector potential which was introduced in the previous paper.

In the special case when the wave incident upon the aperture is plane or spherical, the last term in (1) is found to represent a wave disturbance U(G)(P) which obeys the laws of geometrical optics. The formula (1) then becomes equivalent to a classical result of Maggi and Rubinowicz, which may be regarded as a mathematical refinement of early ideas of Young about the nature of diffraction. It is shown further, that even when the incident wave is not plane or spherical, the last term in (1) is, at least approximately, equal to a “geometrical wave.”

The results suggest a new approach to the solution of certain types of diffraction problems and provide a new insight into the process of diffraction.

© 1962 Optical Society of America

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References

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  1. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
    [Crossref]
  2. G. A. Maggi, Ann. di Mat., IIa,  16, 21 (1888).
  3. A. Rubinowicz, Ann. Physik 53, 257 (1917).
    [Crossref]
  4. T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). For an account of Young’s ideas see A. Rubinowicz, Nature 180, 160 (1957).
    [Crossref]
  5. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959) §8.3.2.
  6. In the special cases when the wave incident on the aperture is plane or spherical, this is true without any approximation.
  7. See, for example, B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Oxford University Press, Oxford, 1950), 2nd. ed., §2.1; or reference 5, §8.9.
  8. The form of the solution given here is the same as that of Rubinowicz. Maggi expressed the solution in a somewhat different form. The equivalence of the results of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923)]. See also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorei der Beugung, (Polska Akademia Nauk, Warsaw, Poland, 1957).
    [Crossref]
  9. A. Rubinowicz, Phys. Rev. 54, 931 (1938).
    [Crossref]
  10. The equation |grad′ϕ| = 1 is the eikonal equation of geometrical optics for a medium of refractive index unity (cf. reference 5, Chap. III).
  11. The analogy between (4.5) and the formulas (3.2), (3.7), and (3.11) for the potential of a plane and a spherical wave should be noted. In fact (3.2), (3.7), and (3.11) follow from (4.5) exactly, on substituting for ϕ the appropriate expressions (ϕ= p · r′for a plane wave and ϕ= ±r′ for a spherical wave).
  12. According to a private communication, the formula (4.5) was also derived by Dr. G. D. Wassermann (Newcastle on Tyne), using a different procedure.
  13. An attempt to extend the early work on the boundary wave to more general fields was also made by R. S. Ingarden, Acta Phys. Polon. 14, 77 (1955).
  14. N. G. van Kampen, Physica 14, 575 (1949). Brief explanation of the principle and references to other papers will be found in reference 5, p. 750.
    [Crossref]
  15. See, for example, R. Courant, Differential and Integral Calculus (Blackie and Son, London, 1942), Vol. I, 2nd ed., p. 214. When AB < 0 the integral is indefinite. However, if the formula (B9) is applied formally one obtains∫0π/2[Acos2φ+Bsin2φ]−1dφ=−πi2|AB|12 and this expression is used in determining the value of Λj when RjRj′< 0.
  16. H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, 1946), p. 474.
  17. J. Focke, Ber. sächs. Ges. Akad. Wiss 101 (1954), Heft 3.
  18. K. Miyamoto, J. Opt. Soc. Am. 48, 57 (1958), Sec. 3.
    [Crossref]
  19. J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
    [Crossref]
  20. To carry out this calculation use first the relation∂(s+ϕ)/∂m=(sˆ+p)·m on the left of (D14). Next choose n,l and m= n× l as base vectors and writesˆ=αn+βl+γm,p=an+bl+cm, where α2+ β2+ γ2= a2+ b2+ c2= 1. The relation (D10) implies that b= −β. Straightforward vector algebra then shows that each term in (D14) is equal to (a− α)/(c− γ).
  21. This contribution is of the same order as the contribution of the second term in the series arising from a critical point of the first kind.

1962 (1)

1958 (1)

1957 (1)

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[Crossref]

1955 (1)

An attempt to extend the early work on the boundary wave to more general fields was also made by R. S. Ingarden, Acta Phys. Polon. 14, 77 (1955).

1954 (1)

J. Focke, Ber. sächs. Ges. Akad. Wiss 101 (1954), Heft 3.

1949 (1)

N. G. van Kampen, Physica 14, 575 (1949). Brief explanation of the principle and references to other papers will be found in reference 5, p. 750.
[Crossref]

1938 (1)

A. Rubinowicz, Phys. Rev. 54, 931 (1938).
[Crossref]

1923 (1)

The form of the solution given here is the same as that of Rubinowicz. Maggi expressed the solution in a somewhat different form. The equivalence of the results of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923)]. See also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorei der Beugung, (Polska Akademia Nauk, Warsaw, Poland, 1957).
[Crossref]

1917 (1)

A. Rubinowicz, Ann. Physik 53, 257 (1917).
[Crossref]

1888 (1)

G. A. Maggi, Ann. di Mat., IIa,  16, 21 (1888).

1802 (1)

T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). For an account of Young’s ideas see A. Rubinowicz, Nature 180, 160 (1957).
[Crossref]

Baker, B. B.

See, for example, B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Oxford University Press, Oxford, 1950), 2nd. ed., §2.1; or reference 5, §8.9.

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959) §8.3.2.

Copson, E. T.

See, for example, B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Oxford University Press, Oxford, 1950), 2nd. ed., §2.1; or reference 5, §8.9.

Courant, R.

See, for example, R. Courant, Differential and Integral Calculus (Blackie and Son, London, 1942), Vol. I, 2nd ed., p. 214. When AB < 0 the integral is indefinite. However, if the formula (B9) is applied formally one obtains∫0π/2[Acos2φ+Bsin2φ]−1dφ=−πi2|AB|12 and this expression is used in determining the value of Λj when RjRj′< 0.

Focke, J.

J. Focke, Ber. sächs. Ges. Akad. Wiss 101 (1954), Heft 3.

Ingarden, R. S.

An attempt to extend the early work on the boundary wave to more general fields was also made by R. S. Ingarden, Acta Phys. Polon. 14, 77 (1955).

Jeffreys, B. S.

H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, 1946), p. 474.

Jeffreys, H.

H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, 1946), p. 474.

Keller, J. B.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[Crossref]

Kottler, F.

The form of the solution given here is the same as that of Rubinowicz. Maggi expressed the solution in a somewhat different form. The equivalence of the results of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923)]. See also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorei der Beugung, (Polska Akademia Nauk, Warsaw, Poland, 1957).
[Crossref]

Lewis, R. M.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[Crossref]

Maggi, G. A.

G. A. Maggi, Ann. di Mat., IIa,  16, 21 (1888).

Miyamoto, K.

Rubinowicz, A.

A. Rubinowicz, Phys. Rev. 54, 931 (1938).
[Crossref]

A. Rubinowicz, Ann. Physik 53, 257 (1917).
[Crossref]

Seckler, B. D.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[Crossref]

van Kampen, N. G.

N. G. van Kampen, Physica 14, 575 (1949). Brief explanation of the principle and references to other papers will be found in reference 5, p. 750.
[Crossref]

Wolf, E.

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
[Crossref]

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959) §8.3.2.

Young, T.

T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). For an account of Young’s ideas see A. Rubinowicz, Nature 180, 160 (1957).
[Crossref]

Acta Phys. Polon. (1)

An attempt to extend the early work on the boundary wave to more general fields was also made by R. S. Ingarden, Acta Phys. Polon. 14, 77 (1955).

Ann. di Mat., IIa (1)

G. A. Maggi, Ann. di Mat., IIa,  16, 21 (1888).

Ann. Physik (2)

A. Rubinowicz, Ann. Physik 53, 257 (1917).
[Crossref]

The form of the solution given here is the same as that of Rubinowicz. Maggi expressed the solution in a somewhat different form. The equivalence of the results of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923)]. See also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorei der Beugung, (Polska Akademia Nauk, Warsaw, Poland, 1957).
[Crossref]

Ber. sächs. Ges. Akad. Wiss (1)

J. Focke, Ber. sächs. Ges. Akad. Wiss 101 (1954), Heft 3.

J. Appl. Phys. (1)

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[Crossref]

J. Opt. Soc. Am. (2)

Phil. Trans. Roy. Soc. (1)

T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). For an account of Young’s ideas see A. Rubinowicz, Nature 180, 160 (1957).
[Crossref]

Phys. Rev. (1)

A. Rubinowicz, Phys. Rev. 54, 931 (1938).
[Crossref]

Physica (1)

N. G. van Kampen, Physica 14, 575 (1949). Brief explanation of the principle and references to other papers will be found in reference 5, p. 750.
[Crossref]

Other (10)

See, for example, R. Courant, Differential and Integral Calculus (Blackie and Son, London, 1942), Vol. I, 2nd ed., p. 214. When AB < 0 the integral is indefinite. However, if the formula (B9) is applied formally one obtains∫0π/2[Acos2φ+Bsin2φ]−1dφ=−πi2|AB|12 and this expression is used in determining the value of Λj when RjRj′< 0.

H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, 1946), p. 474.

To carry out this calculation use first the relation∂(s+ϕ)/∂m=(sˆ+p)·m on the left of (D14). Next choose n,l and m= n× l as base vectors and writesˆ=αn+βl+γm,p=an+bl+cm, where α2+ β2+ γ2= a2+ b2+ c2= 1. The relation (D10) implies that b= −β. Straightforward vector algebra then shows that each term in (D14) is equal to (a− α)/(c− γ).

This contribution is of the same order as the contribution of the second term in the series arising from a critical point of the first kind.

The equation |grad′ϕ| = 1 is the eikonal equation of geometrical optics for a medium of refractive index unity (cf. reference 5, Chap. III).

The analogy between (4.5) and the formulas (3.2), (3.7), and (3.11) for the potential of a plane and a spherical wave should be noted. In fact (3.2), (3.7), and (3.11) follow from (4.5) exactly, on substituting for ϕ the appropriate expressions (ϕ= p · r′for a plane wave and ϕ= ±r′ for a spherical wave).

According to a private communication, the formula (4.5) was also derived by Dr. G. D. Wassermann (Newcastle on Tyne), using a different procedure.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959) §8.3.2.

In the special cases when the wave incident on the aperture is plane or spherical, this is true without any approximation.

See, for example, B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Oxford University Press, Oxford, 1950), 2nd. ed., §2.1; or reference 5, §8.9.

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Figures (8)

Fig. 1
Fig. 1

Illustrating notation relating to Kirchhoff’s integral (2.1).

Fig. 2
Fig. 2

Choice of the surface S.

Fig. 3
Fig. 3

Diffraction of a plane wave at an aperture in an opaque screen.

Fig. 4
Fig. 4

Kirchhoff’s diffraction of a convergent spherical wave. Regions in which the formulas (3.12) and (3.14) apply. The boundaries of the regions are formed by the edge of the geometrical shadow.

Fig. 5
Fig. 5

Illustrating the physical significance of the singularities in the aperture of the vector potential W(Q,P) given by (4.5). The singularities are precisely those aperture points Qj which lie on the geometrical rays that pass through P.

Fig. 6
Fig. 6

A boundary stationary point Qj (critical point of the second kind).

Fig. 7
Fig. 7

A corner of the aperture Qj (critical point of the third kind).

Fig. 8
Fig. 8

Illustrating the notation relating to the evaluation of Λj.

Equations (113)

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U K ( P ) = U ( B ) ( P ) + j F j ( P ) .
U K ( P ) = U ( B ) ( P ) + j F j ( P ) ,
U K ( P ) = U ( B ) ( P ) + U ( G ) ( P ) ,
U K ( P ) = S V ( Q , P ) · n d S ,
V ( Q , P ) = 1 4 π { U ( Q ) grad Q ( exp ( i k s ) s ) ( exp ( i k s ) s ) grad Q U ( Q ) } .
V ( Q , P ) = curl Q W ( Q , P ) ,
W ( Q , P ) = exp ( i k s ) 4 π s s ˆ × 0 exp ( i k μ ) grad U ( r + μ s ˆ ) d μ + W .
U K ( P ) = U ( B ) ( P ) + S F j ( P ) ,
U ( B ) ( P ) = Γ W · l d l ,
F j ( P ) = lim σ j 0 Γ , W · l d l .
U ( P ) = S F j ( P ) + V G j ( P ) .
G j ( P ) = lim ρ j 0 S j V ( Q , P ) · n d S ,
U ( P ) = S F j ( P ) + B F j ( P ) + D F j ( P ) + V G j ( P ) .
U ( P ) = S F j ( P ) + B F j ( P ) .
U K ( P ) = U ( B ) ( P ) + U ( P ) .
U K ( P ) = U ( B ) ( P ) .
U K ( P ) = U ( B ) ( P ) + U ( P ) B F j ( P ) .
U ( P ) = A exp ( i k p · r )
W ( Q , P ) = 1 4 π A exp ( i k p · r ) exp ( i k s ) s s ˆ × p 1 + s ˆ · p ,
U K ( P ) = U ( B ) ( P ) + U ( G ) ( P ) ,
U ( G ) ( P ) = U ( P ) when P is in the direct beam , = 0 when P is in geometrical shadow . }
U ( B ) ( P ) = A 4 π Γ exp ( i k p · r ) exp ( i k s ) s ( s ˆ × p ) · l 1 + s ˆ · p d l ,
U ( P ) = A exp ( i k r ) r ,
W ( Q , P ) = 1 4 π A exp ( i k r ) r exp ( i k s ) s s × r s r + s · r ,
U ( B ) ( P ) = A 4 π Γ exp ( i k r ) r exp ( i k s ) s ( s × r ) · l s r + s · r d l ,
U ( P ) = A exp ( i k r ) r ,
U K ( P ) = U ( B ) ( P ) + S F j ( P ) ,
W ( Q , P ) = 1 4 π A exp ( i k r ) r exp ( i k s ) s s × r s r s · r ,
F j ( P ) = { A exp ( i k r ) / r if P lies in region I A exp ( i k r ) / r if P lies in region II ,
U K ( P ) = U ( B ) ( P ) + U ( G ) ( P ) ,
U ( G ) ( P ) = A exp ( i k r ) / r in region I , = A exp ( i k r ) / r in region II , = 0 in region III }
U ( B ) ( P ) = A 4 π Γ exp ( i k r ) r exp ( i k s ) s × ( s × r ) · l s r s · r d l .
U ( r ) = A ( r ) exp [ i k ϕ ( r ) ] ,
W ( r , r ) = exp ( i k s ) 4 π s s ˆ × ( i k ) 1 grad [ 1 + ( i k ) 1 s ˆ · grad ] U ( r ) .
( i k ) 1 grad U ( r ) = ( grad ϕ + 1 i k grad A A ) U ( r ) .
| grad ϕ ( r ) | = 1 ,
| grad A ( r ) | / | A ( r ) | k = 2 π / λ ,
W ( r , r ) = U ( r ) exp ( i k s ) s 1 4 π s ˆ × grad ϕ ( r ) 1 + s ˆ · grad ϕ ( r ) .
1 4 π s ˆ × grad ϕ ( r ) 1 + s ˆ · grad ϕ ( r ) .
1 + s ˆ · grad ϕ ( r ) = 0.
U ( G ) ( P ) , viz . , U K ( P ) = U ( B ) ( P ) + U ( G ) ( P ) ,
U K ( P ) = U ( B ) ( P ) + S F j ( P ) ,
S F j ( P ) = U ( G ) ( P ) ,
grad ϕ = p ,
U ( B ) ( P ) = Γ U ( Q ) exp ( i k s ) s 1 4 π ( s ˆ × p ) · l 1 + s ˆ · p d l ,
F j ( P ) = lim σ j 0 Γ j U ( Q ) exp ( i k s ) s 1 4 π ( s ˆ × p ) · l 1 + s ˆ · p d l ,
F j ( P ) = U ( Q j ) exp ( i k s j ) Λ j ,
Λ j = ( r j r j R j R j ) 1 2 j ,
j = + 1 if R j R j > 0 , R j > 0 , 1 if R j R j > 0 , R j < 0 , i if R j R j < 0. } .
1 4 π ( s ˆ × p ) · l 1 + s ˆ · p = 1 4 π sin θ 1 + cos θ cos φ .
( / l ) ( s + ϕ ) = 0 ,
U I ( B ) = 1 k 1 2 j U ( Q j ) exp ( i k s j ) 4 π s j [ 2 π | 2 ( s + ϕ ) / l 2 | ] j 1 2 × α j [ ( s ˆ × p ) · l 1 + s ˆ · p ] ,
U II ( B ) = 1 i k j U ( Q j ) exp ( i k s j ) 4 π s j × { sin φ ( p s ˆ ) · n [ ( s ˆ + p ) · l + ] [ ( s ˆ + p ) · l ] } j ,
U K ( P ) = S g ( Q , P ) exp [ i k f ( Q , P ) ] d S ,
g ( Q , P ) = [ ( i k 1 s ) ( s ˆ · n ) ( i k p + grad A A ) · n ] A ( Q ) f ( Q , P ) = s + ϕ ( Q ) ,
S F j = U ( G ) = U k ( I ) .
U I ( B ) = U K ( II ) .
U II ( B ) = U K ( III ) .
U ( r ) = A ( r ) exp [ i k ϕ ( r ) ] .
W ( r , r ) = exp ( i k s ) 4 π s s ˆ × 0 exp ( i k μ ) × grad U ( r + μ s ˆ ) d μ + W = exp ( i k s ) 4 π s s ˆ × 0 ( i k A grad ϕ + grad A ) × exp [ i k ( μ + ϕ ) ] d μ + W .
( / μ ) f ( r + μ s ˆ ) = s ˆ · grad f ( r + μ s ˆ ) .
( / μ ) exp [ i k ( μ + ϕ ) ] = i k ( 1 + s ˆ · grad ϕ ) exp [ i k ( μ + ϕ ) ] .
0 ( i k A grad ϕ + grad A ) exp [ i k ( μ + ϕ ) ] d μ = 0 H ( μ ) μ exp [ i k ( μ + ϕ ) ] d μ = H ( μ ) exp [ i k ( μ + ϕ ) ] | 0 + 0 { s ˆ · grad i k ( 1 + s ˆ · grad ϕ ) } H μ exp i k ( μ + ϕ ) d μ = T 0 H · exp [ i k ( μ + ϕ ) ] | 0 + ( 1 / i k ) T H · exp i k ( μ + ϕ ) | 0 + ( 1 i k ) 2 0 T 2 H · μ exp [ i k ( μ + ϕ ) ] d μ ,
H = A grad ϕ + ( i k ) 1 grad A 1 + s ˆ · grad ϕ ,
T = 1 1 + s ˆ · grad ϕ s ˆ · grad .
W ( r , r ) = exp { i k ( s + ϕ ( r ) ] } 4 π s s ˆ n = 0 1 ( i k ) n { s ˆ · grad 1 + s ˆ · grad ϕ ( r ) } n { A ( r ) grad ϕ ( r ) + ( i k ) 1 grad A ( r ) 1 + s ˆ · grad ϕ ( r ) } .
W ( r , r ) = exp ( i k s ) 4 π s U ( r ) s ˆ × n = 0 1 ( i k ) n B n ( r , s ˆ ) .
B 0 = p 1 + s ˆ · p , B 1 = 1 A ( r ) grad A ( r ) 1 + s ˆ · p + 1 A ( r ) ( s ˆ · grad 1 + s ˆ · p ) A ( r ) p 1 + s ˆ · p , }
p = grad ϕ ( r ) .
W ( Q , P ) = 1 4 π U ( Q ) exp ( i k s ) s s ˆ × p 1 + s ˆ · p ,
p = grad ϕ ( Q ) .
F j = U ( Q j ) exp ( i k s j ) · Λ j ,
Λ j = 1 4 π s j lim σ j 0 Γ j ( s ˆ × p ) · l 1 + s ˆ · p d l .
Λ j = 1 2 π s j lim σ j 0 0 2 π σ j θ j ( φ ) d φ .
θ j = σ j [ 1 s j 1 ρ j ( φ ) ] ,
1 ρ j ( φ ) = cos 2 φ r j + sin 2 φ r j .
Λ j = 1 2 π s j 0 2 π [ 1 s j 1 ρ j ( φ ) ] 1 d φ = 1 2 π s j 0 2 π [ ( 1 s j 1 r j ) cos 2 φ + ( 1 s j 1 r j ) sin 2 φ ] 1 d φ .
[ A cos 2 φ + B sin 2 φ ] 1 d φ = 1 ( A B ) 1 2 tan 1 [ ( A B ) 1 2 A tan φ ] ,
Λ j = 1 s j | 1 s j 1 r j | 1 2 | 1 s j 1 r j | 1 2 j = | r j r j R j R j | 1 2 j ,
j = + 1 when R j R j > 0 , R j > 0 , = 1 when R j R j > 0 , R j < 0 , = i when R j R j < 0. }
F j = U ( Q j ) exp ( i k s j ) | r j r j R j R j | 1 2 j .
U ( B ) = 1 4 π Γ U ( Q ) exp ( i k s ) s ( s ˆ × p ) · l 1 + s ˆ · p d l
J ( k ) = a b g ( l ) exp [ i k f ( l ) ] d l ,
J I ( k ) = ( 2 π k ) 1 2 j α j | f ( l j ) | 1 2 g ( l j ) exp [ i k f ( l j ) ] ,
α j = exp ( i π / 4 ) if f ( l j ) > 0 , = exp ( i π / 4 ) if f ( l j ) < 0. }
J II ( k ) = 1 i k { g ( b ) f ( b ) exp [ i k f ( b ) ] g ( a ) f ( a ) exp [ i k f ( a ) ] } .
U = A e i k ϕ .
( s + ϕ ) / l = ( s ˆ + p ) · l = 0.
U I ( B ) ( P ) = 1 k 1 2 j U ( Q j ) exp ( i k s j ) 4 π s j × { 2 π | 2 ( s + ϕ ) / l 2 | } 1 2 α j [ ( s ˆ × p ) · l 1 + s ˆ · p ] j ,
U II ( P ) = 1 i k j U ( Q j ) exp ( i k s j ) 4 π s j × [ s ˆ × p 1 + s ˆ · p ] j [ l ( s + ϕ ) / l ] Q j + Q j .
[ s ˆ × p ] 1 + s ˆ · p · { [ ( s ˆ + p ) · l + ] l [ ( s ˆ + p ) · l ] l + } [ ( s ˆ + p ) · l + ] [ ( s ˆ + p ) · l ] = s ˆ × p 1 + s ˆ · p · ( s ˆ + p ) × ( l + × l ) [ ( s ˆ + p ) · l + ] [ ( s ˆ + p ) · l ] = s ˆ × p 1 + s ˆ · p · ( s ˆ + p ) × n sin φ [ ( s ˆ + p ) · l + ] [ ( s ˆ + p ) · l ] .
( s ˆ × p ) · [ ( s ˆ + p ) × n ] = { p × [ ( s ˆ + p ) × n ] } · s ˆ = { ( s ˆ + p ) ( p · n ) n ( p · s ˆ + 1 ) } · s ˆ = ( 1 + s ˆ · p ) ( p s ˆ ) · n
U II ( P ) = 1 i k j U ( Q j ) exp ( i k s j ) 4 π s j × { ( p s ˆ ) · n sin φ [ ( s ˆ + p ) · l + ] [ ( s ˆ + p ) · l ] } j .
U k ( P ) = 1 4 π S { ( i k 1 s ) ( s ˆ . n ) ( i k p + grad log A ) . n } × A exp [ i k ( s + ϕ ) ] s d ξ d η ,
U K ( P ) = i k 4 π S ( s ˆ p ) · n A s exp [ i k ( s + ϕ ) ] d ξ d η .
( s + ϕ ) ξ = ( s + ϕ ) η = 0 , i . e . , s ˆ + p = 0.
U K ( P ) = 2 π i k j j ( | Δ j | ) 1 2 × { i k ( s ˆ p ) · n 4 π s A exp [ i k ( s + ϕ ) ] } j ,
Δ j = { 2 ( s + ϕ ) ξ 2 2 ( s + ϕ ) η 2 [ 2 ( s + ϕ ) ξ η ] 2 } j ,
j = + 1 if Δ j > 0 , 2 ( s + ϕ ) / ξ 2 > 0 , = 1 if Δ j > 0 , 2 ( s + ϕ ) / ξ 2 < 0 , = i if Δ j < 0 , }
U K ( I ) ( P ) = j U ( Q j ) exp ( i k s j ) ( p · n ) j s j ( | Δ j | ) 1 2 j .
p · n s j ( | Δ j | ) 1 2 = ( d σ Q d σ P ) 1 2 ,
d σ / d σ P = | r j r j / R j R j |
U K ( I ) ( P ) = j U ( Q j ) exp ( i k s j ) | r j r j R j R j | 1 2 j
( s + ϕ ) / l = 0 , i . e . , ( s ˆ + p ) · l = 0.
U K ( II ) ( P ) = i i k [ 2 π k | 2 ( s + ϕ ) / l 2 | ] { i k 4 π ( s ˆ p ) · n × A s exp [ i k ( s + ϕ ) ] } j { α j ( s + ϕ ) / m } j = 1 k 1 2 j U ( Q j ) exp ( i k s j ) 4 π s j [ 2 π | 2 ( s + ϕ ) / l 2 | ] 1 2 × [ n · ( p s ˆ ) ( s + ϕ ) / m ] j α j ,
α = { exp ( + i π / 4 ) when [ 2 ( s + ϕ ) / l 2 ] j > 0 , exp ( i π / 4 ) when [ 2 ( s + ϕ ) / l 2 ] j < 0 ,
m = n × l ,
[ n · ( p s ˆ ) ( s + ϕ ) / m ] j = [ ( s ˆ × p ) · l 1 + s ˆ · p ] j .
U K ( III ) ( P ) = 1 k i { sin φ k [ ( s + ϕ ) / l ] + [ ( s + ϕ ) / l ] } j × { i k 4 π ( s ˆ p ) · n A s exp [ i k ( s + ϕ ) ] } j ,
U K ( III ) ( P ) = 1 i k i U ( Q j ) exp ( i k s j ) 4 π s j × { sin φ . ( p s ˆ ) . n [ ( s ˆ + p ) · l + ] [ ( s ˆ + p ) · l ] } j .
0π/2[Acos2φ+Bsin2φ]1dφ=πi2|AB|12
(s+ϕ)/m=(sˆ+p)·m
sˆ=αn+βl+γm,p=an+bl+cm,